Update-Efficiency and Local Repairability Limits for Capacity Approaching Codes

TL;DR

This paper explores the limits of efficient update and local repair in capacity-approaching codes for distributed storage, establishing bounds on how many bits must change or be accessed for updates and repairs.

Contribution

It provides new theoretical bounds on update-efficiency and local repairability for capacity-achieving codes, including conditions for optimal error correction and repair.

Findings

Update changes scale logarithmically with block-length for nontrivial rates.

Maximum codeword symbols needed for repair scale as log(1/epsilon).

Existence of capacity-achieving codes meeting these bounds.

Abstract

Motivated by distributed storage applications, we investigate the degree to which capacity achieving encodings can be efficiently updated when a single information bit changes, and the degree to which such encodings can be efficiently (i.e., locally) repaired when single encoded bit is lost. Specifically, we first develop conditions under which optimum error-correction and update-efficiency are possible, and establish that the number of encoded bits that must change in response to a change in a single information bit must scale logarithmically in the block-length of the code if we are to achieve any nontrivial rate with vanishing probability of error over the binary erasure or binary symmetric channels. Moreover, we show there exist capacity-achieving codes with this scaling. With respect to local repairability, we develop tight upper and lower bounds on the number of remaining encoded bits that are needed to recover a single lost bit of the encoding. In particular, we show that if the code-rate is $\epsilon$ less than the capacity, then for optimal codes, the maximum number of codeword symbols required to recover one lost symbol must scale as $\log1/\epsilon$. Several variations on---and extensions of---these results are also developed.